A new simple proof for Lum-Chua's conjecture

TL;DR

This paper presents a concise and straightforward proof of Lum-Chua's conjecture, demonstrating that a specific class of planar piecewise linear systems can have at most one limit cycle, and characterizes its stability.

Contribution

It introduces a novel characterization for Poincaré half-maps that simplifies the proof and reveals new properties of the limit cycle, including hyperbolicity and stability criteria.

Findings

At most one limit cycle exists in the system.

If a limit cycle exists, it is hyperbolic.

The stability of the limit cycle is determined by a simple parameter relationship.

Abstract

The already proved Lum-Chua's conjecture says that a continuous planar piecewise linear differential system with two zones separated by a straight line has at most one limit cycle. In this paper, we provide a new proof by using a novel characterization for Poincar\'e half-maps in planar linear systems. This proof is very short and straightforward, because this characterization avoids the inherent flaws of the usual methods to study piecewise linear systems (the appearance of large case-by-case analysis due to the unnecessary discrimination between the different spectra of the involved matrices, to deal with transcendental equations forced by the implicit occurrence of flight time, ...). In addition, the application of the characterization allow us to prove that if a limit cycle exists, then it is hyperbolic and its stability is determined by a simple relationship between the parameters. To the best of our knowledge, the hyperbolicity of the limit cycle and this simple expression for its stability have not been pointed out before.